Ég finn ekki skrúfjárnið mitt, svo ég spyr hana hvort hún eigi annað verkfæri.

skrúfjárn means screwdriver (literally something like “screw-iron/tool”).
skrúfjárnið is the definite form: the screwdriver. Icelandic commonly attaches the definite article to the end of the noun:

• skrúfjárn = a screwdriver
• skrúfjárnið = the screwdriver

The ending -ið is the definite ending for many neuter nouns in the singular nominative/accusative.

In Icelandic, possessives very often come after a definite noun:

• skrúfjárnið mitt = my screwdriver (literally “the screwdriver my”)

You can also put the possessive before the noun, but then the noun is usually indefinite and the structure changes:

• mitt skrúfjárn = my screwdriver (more like “a screwdriver of mine” / less “the specific one”)

The possessive must agree with the noun in gender, number, and case.
skrúfjárn is neuter singular, and here it’s the object of finn (accusative; but neuter singular looks the same in nominative/accusative), so you get:

• masculine: minn
• feminine: mín
• neuter: mitt ✅

Here svo means so / therefore / and so linking the two clauses:

• ..., svo ... = “..., so ...”

Other common options (with slightly different tone/structure) include:

• þannig að = “so that / so” (often a bit more explicit)
• þess vegna = “therefore / because of that” (more “therefore” than “so”)

hana is accusative (feminine singular) of hún.
The verb spyrja commonly takes the person you ask in the accusative:

• að spyrja einhvern (ACC) = to ask someone
  So: ég spyr hana = “I ask her”.

Not necessarily. Icelandic has both patterns depending on what follows:

• spyrja einhvern um eitthvað = ask someone about something
• spyrja einhvern hvort / hvað / hvenær ... = ask someone whether/what/when ... (no um needed)

Your sentence uses an indirect yes/no question (hvort-clause), so um is not used.

hvort introduces an indirect yes/no question and means whether/if (in the sense of “whether,” not conditional “if”):

• Ég spyr hana hvort ... = “I ask her whether ...”

In Icelandic, hvort is very common in reported questions like this.

á is the present indicative form (“she has/owns” in a straightforward main clause): hún á.
In indirect questions (especially after hvort), Icelandic often uses the subjunctive: hún eigi. This is a very typical, standard choice in careful Icelandic.

So:

• direct: Á hún annað verkfæri? = “Does she have another tool?”
• indirect: ... hvort hún eigi annað verkfæri. = “... whether she has another tool.”

In the main clause, Icelandic often has a V2-style order, but in a subordinate clause (like one introduced by hvort), the word order is typically:

• conjunction – subject – verb – ...
  So: hvort hún eigi ... is the normal subordinate-clause order.

annað means another and it must agree with the noun it describes.
verkfæri is neuter singular, so you get the neuter form annað:

• masculine: annar
• feminine: önnur
• neuter: annað ✅